Worst-case Throughput Analysis for Parametric Rate and Parametric - PowerPoint PPT Presentation

Introduction Preliminaries Performance analysis for PSADF Experiments Worst-case Throughput Analysis for Parametric Rate and Parametric Actor Execution Time Scenario-Aware Dataflow Graphs Mladen Skelin Norwegian University of Science and Technology Department of Engineering Cybernetics SynCoP 2014 Mladen Skelin

Introduction Preliminaries Performance analysis for PSADF Experiments Outline I Introduction 1 Preliminaries 2 SDF SADF Performance analysis for PSADF 3 Experiments 4 Mladen Skelin

Introduction Preliminaries Performance analysis for PSADF Experiments Performance analysis Throughput Latency Mladen Skelin

Introduction Preliminaries Performance analysis for PSADF Experiments Dataflow models and dynamic applications Synchronous dataflow 1 (SDF) as a restriction of Kahn process networks SDF is very fitted for modeling regular streaming applications . . . , but not dynamic applications Therefore we need more expressive models than SDF An example SDFG 1 Lee, E.A. et al. : Synchronous data flow. Mladen Skelin

Introduction Preliminaries Performance analysis for PSADF Experiments Scenario-aware dataflow (SADF) SADF 1 encodes the dynamism of an application by identifying a FINITE number of different behaviours called modes or scenario where every scenario is given by a an SDFG A finite state machine (FSM) is used to encode scenario occurrence patterns SADF is equiped with a technique that yields the exact performance guarantees 2 An example SADFG 1 Theelen, B.D. et al. : A scenario-aware data flow model for combined long-run average and worst-case performance analysis. 2 Geilen, M. et al. : Worst-case Performance Analysis of Synchronous Dataflow Scenarios. Mladen Skelin

Introduction Preliminaries Performance analysis for PSADF Experiments SADF SADF is limited by its finiteness: Example application ProcessData.A(out g, out h); for (i=0; i<=g; i++){ for (j=0; j<=h; j++){ // Perform two tasks in parallel #region ParallelTasks // Perform two tasks in parallel Parallel.Invoke(() => { ProcessData.B(i,j); }, // close first parallel action () => { ProcessData.C(i,j); } // close second parallel action ); // close Parallel.Invoke #endregion ProcessData.D(i,j); } } Mladen Skelin

Introduction Preliminaries Performance analysis for PSADF Experiments SADF � 0 , n � g takes a value from the interval 2 0 , m � � h take a value from the interval 2 Application exhibits as many behaviours as there are integer points in the rational 2-polytope P n , m given by the set of constraints { 0 ≤ 1 2 n , 0 ≤ 1 2 m } For n = 4500 and m = 2001 we would have to generate 2 , 252 , 126 SDFGs Mladen Skelin

Introduction Preliminaries Performance analysis for PSADF Experiments Parmetrization of SADF Parametrization as a solution to the problem of SADF “finitness” We raise the problem of SADF parametrization in the scope of existing parametric dataflow models Schedulable parametric dataflow 1 (SPDF) provides a high level of generalization SPDF and syntactical convenience Goal is to embed SPDF into SADF for the purpose of performance analysis 1 Fradet, P . et al. : SPDF: A schedulable parametric data-flow MoC. Mladen Skelin

Introduction Preliminaries Performance analysis for PSADF Experiments Parmetrization of SADF We define the parametric rate and parametric execution time SADF (PSADF) We formally define the semantics of the PSADF model relevant for throughput analysis based on (max,+) linear system theory and (max,+) automata We give the algorithms for worst-case throughput analysis of PSADF graphs with a fully connected, possibly infinite state transition system Mladen Skelin

Introduction Preliminaries SDF Performance analysis for PSADF SADF Experiments Outline I Introduction 1 Preliminaries 2 SDF SADF Performance analysis for PSADF 3 Experiments 4 Mladen Skelin

Introduction Preliminaries SDF Performance analysis for PSADF SADF Experiments SDF Definition A Synchronous dataflow graph (SDFG) is a tuple SDFG = ( G , R , D , i , r , e ) , where: G is a connected directed graph ( A , E ) with A a set of nodes ( actors ) and E ⊆ A × A a set of edges ( channels ); R ⊂ N + is a set of rates used to define SDFG rates by the grammar FR :: = r , where r ∈ R ; D ⊂ R + 0 is a set of actor execution times used to define SDFG actor execution times by the grammar FD :: = d , where d ∈ D ; i : E → N 0 returns for each channel its number of initial tokens; r : A × E → FR returns for each port (represented by an actor and one of its channels) its rate; e : A → FD returns for each actor its execution time. Mladen Skelin

Introduction Preliminaries SDF Performance analysis for PSADF SADF Experiments (max,+) algebra for SDF Let � γ denote the vector of production times of tokens that exist in their different channels in between iterations The evolution of the graph is then given by the following equation: � γ k + 1 = G � γ k , where G = { g ij } is a characteristic (max,+) matrix of the graph Entry g ij specifies the minimal elapsed time from the production time of the j th token in the previous iteration to the production time of the i th in the current iteration Matrix G defines a corresponding (max,+) automaton graph (MPAG) Throughput of the SDFG takes the value 1 / λ , where λ is the maximum cycle mean (MCM) of the MPAG Mladen Skelin

Introduction Preliminaries SDF Performance analysis for PSADF SADF Experiments (max,+) algebra for SDF MPAG of the An example SDFG example SDFG − ∞ − ∞ − ∞ ⎡ 29 29 ⎤ − ∞ − ∞ 33 4 33 ⎢ ⎥ G = 63 − ∞ 30 63 − ∞ ⎢ ⎥ ⎢ ⎥ − ∞ − ∞ − ∞ − ∞ 0 ⎣ ⎦ − ∞ 64 5 31 64 Mladen Skelin

Introduction Preliminaries SDF Performance analysis for PSADF SADF Experiments (max,+) algebra for SDF For example, � γ 1 can be calculated as: − ∞ − ∞ − ∞ max ( 29 + 0 , 29 + 0 ) ⎡ 29 29 ⎤ ⎡ 0 ⎤ ⎡ ⎤ ⎡ 29 ⎤ − ∞ − ∞ max ( 33 + 0 , 4 + 0 , 33 + 0 ) 33 4 33 0 33 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ γ 1 = � − ∞ − ∞ ⎦ = max ( 63 + 0 , 30 + 0 , 63 + 0 ) ⎦ = ⎢ 63 30 63 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 63 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ − ∞ − ∞ − ∞ − ∞ max ( 0 + 0 ) 0 0 0 ⎣ ⎦ ⎣ ⎣ ⎣ ⎦ − ∞ max ( 64 + 0 , 5 + 0 , 31 + 0 , 64 + 0 ) 64 5 31 64 0 64 Mladen Skelin

Introduction Preliminaries SDF Performance analysis for PSADF SADF Experiments SADF Definition A Scenario-aware dataflow graph (SADFG) is a tuple SADFG = ( S , F ) , where: S = { s i | s i = ( scen i , G i ) } is a set of ordered pairs of scenarios and their corresponding SDFGs; F = ( Q , q 0 , δ , Σ , E ) is the scenario finite state machine consisting of a finite set Q of states, an initial state q 0 ∈ Q , a transition relation δ ⊆ Q × Q , a scenario labelling Σ : Q → S and a set of final states E , where E = Q . Mladen Skelin

Introduction Preliminaries SDF Performance analysis for PSADF SADF Experiments (max,+) algebra for SADF Let G ( s i ) denote the n × n (max,+) characteristic matrix for the scenario s i The completion time of a k -long sequence of scenarios can then be defined as a sequence of (max,+) matrix multiplications � A ( s 1 ... s k ) = G ( s k ) ... G ( s 1 ) 0 The worst case increase of A ( s ) for a growing length of s specifies the worst-case throughput for any sequence of scenarios Again, the inverse of the MCM ( 1 / λ ) of the SADF MPAG denotes the worst-case throughput A special case that arises in practice is when scenarios can occur in arbitrary order, yielding the SADF FSM to be fully connected and with a single state for each scenario Then is the throughput of an SADF equal to the maximum cycle mean of the MPAG that corresponds to the (max,+) matrix G = max q ∈ Q ( G ( Σ ( q ))) Mladen Skelin

Introduction Preliminaries SDF Performance analysis for PSADF SADF Experiments (max,+) algebra for SADF MPAG of the example An example SADFG SADFG 29 − ∞ − ∞ 29 − ∞ 28 − ∞ − ∞ 28 − ∞ ⎡ ⎤ ⎡ ⎤ 33 4 − ∞ 33 − ∞ 34 6 − ∞ 34 − ∞ ⎢ ⎥ ⎢ ⎥ G ( a ) = 63 − ∞ 30 63 − ∞ G ( b ) = 72 − ∞ 24 72 − ∞ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ − ∞ − ∞ − ∞ − ∞ − ∞ − ∞ − ∞ − ∞ 0 0 ⎣ ⎦ ⎣ ⎦ 64 5 31 64 − ∞ 82 16 34 82 − ∞ Mladen Skelin

Introduction Preliminaries Performance analysis for PSADF Experiments Outline I Introduction 1 Preliminaries 2 SDF SADF Performance analysis for PSADF 3 Experiments 4 Mladen Skelin

Introduction Preliminaries Performance analysis for PSADF Experiments Parametrization of SADF SADF becomes impractical or even infeasible when it faces applications with a vast set of possible behaviours We need a compact representation Parametrization is not trivial (liveness, boundedness and schedulability) Parametrization of SADF by embedding an parametric model into SADF SPDF – liveness and boundedness properties for SPDF are decidable SPDF extends SDF by allowing rates to be parametric while preserving static schedulability We add the notion of time to SPDF Mladen Skelin